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arXiv:1410.0169 · Physical Review A

Energy expectation values of a particle in nonstationary fields

Alexander J. Silenko

Published Jan 20, 2015• 33 citations

Abstract

We show that the origin of the nonequivalence of Hamiltonians in different representations is a change of the form of the time-derivative operator at a time-dependent unitary transformation. This nonequivalence does not lead to an ambiguity of the energy expectation values of a particle in nonstationary fields but assigns the basic representation. It has been explicitly or implicitly supposed in previous investigations that this representation is the Dirac one. We prove the alternative assertion about the basic role of the Foldy-Wouthuysen representation. We also derive the general equation for the energy expectation values in the Dirac representation. As an example, we consider a spin-1/2 particle with anomalous magnetic and electric dipole moments in strong time-dependent electromagnetic fields. We apply the obtained results to a spin-1/2 particle in a plane monochromatic electromagnetic wave and give an example of the exact Foldy-Wouthuysen transformation in the nonstationary case.

DOI: 10.1103/physreva.91.012111

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We show that the origin of the nonequivalence of Hamiltonians in different representations is a change of the form of the time-derivative operator at a time-dependent unitary transformation. This nonequivalence does not lead to an ambiguity of the energy expectation values of a particle in nonstationary fields but assigns the basic representation. It has been explicitly or implicitly supposed in previous investigations that this representation is the Dirac one. We prove the alternative assertion about the basic role of the Foldy-Wouthuysen representation. We also derive the general equation for the energy expectation values in the Dirac representation. As an example, we consider a spin-1/2 particle with anomalous magnetic and electric dipole moments in strong time-dependent electromagnetic fields. We apply the obtained results to a spin-1/2 particle in a plane monochromatic electromagnetic wave and give an example of the exact Foldy-Wouthuysen transformation in the nonstationary case.

The Dirac equation for a charged particle in a time-dependent electromagnetic field is reduced to the Pauli equation in the non-relativistic limit by making a time-dependent Foldy-Wouthuysen transformation in a completely gauge-invariant way. In a time-dependent field, the expectation value of the Hamiltonian is dependent on the gauge, and so it cannot be the energy. The energy is obtained by taking the expectation value of the energy operator, the time rate of change of which is the average of the quantum mechanical power operator. The Hamiltonian formulation of a charged particle in a time-dependent external electromagnetic field is satisfactory when this distinction between the Hamiltonian, which describes the time evolution of the system, and the energy operator, which is gauge invariant, is made.

Although the time-dependent Foldy-Wouthuysen (FW) transformation of the "Schr\"odinger" equation $\frac{i\ensuremath{\partial}\ensuremath{\psi}}{\ensuremath{\partial}t}=H\ensuremath{\psi}$ yields the transformed equation $\frac{i\ensuremath{\partial}{\ensuremath{\psi}}^{\ensuremath{'}}}{\ensuremath{\partial}t}={H}^{\ensuremath{'}}{\ensuremath{\psi}}^{\ensuremath{'}}$, where ${\ensuremath{\psi}}^{\ensuremath{'}}=U\ensuremath{\psi}$ and ${H}^{\ensuremath{'}}=UH{U}^{\ensuremath{-}1}\ensuremath{-}\frac{\mathrm{iU}\ensuremath{\partial}({U}^{\ensuremath{-}1})}{\ensuremath{\partial}t}$, the expectation values (${\ensuremath{\psi}}_{1}, H{\ensuremath{\psi}}_{j}$) of

H

are not equal to (${{\ensuremath{\psi}}_{i}}^{\ensuremath{'}}, {H}^{\ensuremath{'}}{{\ensuremath{\psi}}_{j}}^{\ensuremath{'}}$), but rather, to (${{\ensuremath{\psi}}_{i}}^{\ensuremath{'}}, \mathrm{UH}{U}^{\ensuremath{-}1}{{\ensuremath{\psi}}_{j}}^{\ensuremath{'}}$). But one still has $({{\ensuremath{\psi}}_{i}}^{\ensuremath{'}}, {{\ensuremath{\psi}}_{j}}^{\ensuremath{'}})=({\ensuremath{\psi}}_{i}, {\ensuremath{\psi}}_{j})$. I discuss implications for perturbation calculations, scattering amplitudes, electrodynamics, and the external-field $\ensuremath{\pi}N$ ambiguity, and look at exact special cases.

The Foldy-Wouthuysen transformation for relativistic spin-1 particles interacting with nonuniform electric and uniform magnetic fields is performed. The Hamilton operator in the Foldy-Wouthuysen representation is determined. It agrees with the Lagrangian obtained by Pomeransky, Khriplovich, and Sen'kov. The classical and quantum formulae for the Hamiltonian agree. The validity of the Corben-Schwinger equations is confirmed. However, it is difficult to generalize these equations in order to take into account the quadrupole moment defined by a particle charge distribution. The known second-order wave equations are not quite satisfactory because they contain non-Hermitian terms. The Hermitian second-order wave equation is derived.